Question

The angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{5\pi }{6}$ and the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$ is $\frac{-6}{\sqrt{3}},$ then $|\overrightarrow{a}|$ is equal to:

• A. 6
• B. $\frac{\sqrt{3}}{2}$
• C. 12
• D. 4
• E. 16

Answer 1

Answer: (D)

$\because$ $\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \frac{5\pi }{6}$
$=-\frac{|\overrightarrow{a}||\overrightarrow{b}|\sqrt{3}}{2}$
Since, the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$ $\therefore$ $-\frac{6}{\sqrt{3}}=-\frac{|\overrightarrow{a}||\overrightarrow{b}|\sqrt{3}}{2|\overrightarrow{b}|}$
$\Rightarrow$ $|\overrightarrow{a}|=\frac{6\times 2}{3}=4$